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### e: The Story of a NumberEli Maor

COPYRIGHT NOTICE: Published by Princeton University Press and copyrighted, © 1994, by Princeton University Press. All rights reserved. No part of this book may be reproduced in any form by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from the publisher, except for reading and browsing via the World Wide Web. Users are not permitted to mount this file on any network servers. Follow links for Class Use and other Permissions. For more information, send e-mail to permissions@press.princeton.edu

### Preface

It must have been at the age of nine or ten when I first encountered the number pi. My father had a friend who owned a workshop, and one day I was invited to visit the place. The room was filled with tools and machines, and a heavy oily smell hung over the place. Hardware had never particularly interested me, and the owner must have sensed my boredom when he took me aside to one of the bigger machines that had several flywheels attached to it. He explained that no matter how large or small a wheel is, there is always a fixed ratio between its circumference and its diameter, and this ratio is about 3 1/7. I was intrigued by this strange number, and my amazement was heightened when my host added that no one had yet written this number exactly--one could only approximate it. Yet so important is this number that a special symbol has been given to it, the Greek letter pi. Why, I asked myself, would a shape as simple as a circle have such a strange number associated with it? Little did I know that the very same number had intrigued scientists for nearly four thousand years, and that some questions about it have not been answered even today.

Several years later, as a high school junior studying algebra, I became intrigued by a second strange number. The study of logarithms was an important part of the curriculum, and in those days--well before the appearance of hand-held calculators--the use of logarithmic tables was a must for anyone wishing to study higher mathematics. How dreaded were these tables, with their green cover, issued by the Israeli Ministry of Education! You got bored to death doing hundreds of drill exercises and hoping that you didn't skip a row or look up the wrong column. The logarithms we used were called "common"--they used the base 10, quite naturally. But the tables also had a page called "natural logarithms." When I inquired how anything can be more "natural" than logarithms to the base 10, my teacher answered that there is a special number, denoted by the letter e and approximately equal to 2.71828, that is used as a base in "higher" mathematics. Why this strange number? I had to wait until my senior year, when we took up the calculus, to find out.

In the meantime pi had a cousin of sorts, and a comparison between the two was inevitable--all the more so since their values are so close. It took me a few more years of university studies to learn that the two cousins are indeed closely related and that their relationship is all the more mysterious by the presence of a third symbol, i, the celebrated "imaginary unit," the square root of -1. So here were all the elements of a mathematical drama waiting to be told.

The story of pi has been extensively told, no doubt because its history goes back to ancient times, but also because much of it can be grasped without a knowledge of advanced mathematics. Perhaps no book did better than Petr Beckmann's A History of pi, a model of popular yet clear and precise exposition. The number e fared less well. Not only is it of more modern vintage, but its history is closely associated with the calculus, the subject that is traditionally regarded as the gate to "higher" mathematics. To the best of my knowledge, a book on the history of e comparable to Beckmann's has not yet appeared. I hope that the present book will fill this gap. My goal is to tell the story of e on a level accessible to readers with only a modest background in mathematics. I have minimized the use of mathematics in the text itself, delegating several proofs and derivations to the appendixes. Also, I have allowed myself to digress from the main subject on occasion to explore some side issues of historical interest. These include biographical sketches of the many figures who played a role in the history of e, some of whom are rarely mentioned in textbooks. Above all, I want to show the great variety of phenomena--from physics and biology to art and music--that are related to the exponential function ex, making it a subject of interest in fields well beyond mathematics.

On several occasions I have departed from the traditional way that certain topics are presented in calculus textbooks. For example, in showing that the function y = ex is equal to its own derivative, most textbooks first derive the formula d(1n x)/dx = l/x, a long process in itself. Only then, after invoking the rule for the derivative of the inverse function, is the desired result obtained. I have always felt that this is an unnecessarily long process: one can derive the formula d(ex)/dx = ex directly--and much faster--by showing that the derivative of the general exponential function y = bx is proportional to bx and then finding the value of b for which the proportionality constant is equal to 1 (this derivation is given in Appendix 4). For the expression cos x + i sin x, which appears so frequently in higher mathematics, I have used the concise notation cis x (pronounced "ciss x"), with the hope that this much shorter notation will be used more often. When considering the analogies between the circular and the hyperbolic functions, one of the most beautiful results, discovered around 1750 by Vincenzo Riccati, is that for both types of functions the independent variable can be interpreted geometrically as an area, making the formal similarities between the two types of functions even more striking. This fact--seldom mentioned in the textbooks--is discussed in Chapter 12 and again in Appendix 7.

In the course of my research, one fact became immediately clear: the number e was known to mathematicians at least half a century before the invention of the calculus (it is already referred to in Edward Wright's English translation of John Napier's work on logarithms, published in 1618). How could this be? One possible explanation is that the number e first appeared in connection with the formula for compound interest. Someone--we don't know who or when--must have noticed the curious fact that if a principal P is compounded n times a year for t years at an annual interest rate r, and if n is allowed to increase without bound, the amount of money S, as found from the formula S = P(1 + r/n)nt, seems to approach a certain limit. This limit, for P = 1, r = 1, and t = 1, is about 2.718. This discovery--most likely an experimental observation rather than the result of rigorous mathematical deduction--must have startled mathematicians of the early seventeenth century, to whom the limit concept was not yet known. Thus, the very origins of the the number e and the exponential function ex may well be found in a mundane problem: the way money grows with time. We shall see, however, that other questions--notably the area under the hyperbola y = l/x--led independently to the same number, leaving the exact origin of e shrouded in mystery. The much more familiar role of e as the "natural" base of logarithms had to wait until Leonhard Euler's work in the first half of the eighteenth century gave the exponential function the central role it plays in the calculus.

I have made every attempt to provide names and dates as accurately as possible, although the sources often give conflicting information, particularly on the priority of certain discoveries. The early seventeenth century was a period of unprecedented mathematical activity, and often several scientists, unaware of each other's work, developed similar ideas and arrived at similar results around the same time. The practice of publishing one's results in a scientific journal was not yet widely known, so some of the greatest discoveries of the time were communicated to the world in the form of letters, pamphlets, or books in limited circulation, making it difficult to determine who first found this fact or that. This unfortunate state of affairs reached a climax in the bitter priority dispute over the invention of the calculus, an event that pitted some of the best minds of the time against one another and was in no small measure responsible for the slowdown of mathematics in England for nearly a century after Newton.

As one who has taught mathematics at all levels of university instruction, I am well aware of the negative attitude of so many students toward the subject. There are many reasons for this, one of them no doubt being the esoteric, dry way in which we teach the subject. We tend to overwhelm our students with formulas, definitions, theorems, and proofs, but we seldom mention the historical evolution of these facts, leaving the impression that these facts were handed to us, like the Ten Commandments, by some divine authority. The history of mathematics is a good way to correct these impressions. In my classes I always try to interject some morsels of mathematical history or vignettes of the persons whose names are associated with the formulas and theorems. The present book derives partially from this approach. I hope it will fulfill its intended goal.

Many thanks go to my wife, Dalia, for her invaluable help and support in getting this book written, and to my son Eyal for drawing the illustrations. Without them this book would never have become a reality.

Skokie, Illinois
January 7, 1993